The intersection-cohomology ring conjecture for nonabelian hypertoric quotients

Let $G$ be a group with maximal torus $T$ and Weyl group $W$, and let $\mathfrak{M}_{\alpha,0}(G)$ and $\mathfrak{M}_{\alpha,0}(T)$ be the corresponding symplectic quotients. Assume that these spaces are smooth for generic $(\alpha,0)$. Let $\mathbf{IC}(\mathfrak{M}_{0,0}(G))$ denote the intersection cohomology sheaf, let $I\!H^*$ denote intersection cohomology, and let $e_0$ be the product of the roots as above. The intersection-cohomology ring conjecture. Suppose that $\mathfrak{M}_{\alpha,0}(G)$ and $\mathfrak{M}_{\alpha,0}(T)$ are smooth for generic $(\alpha,0)$. Then the intersection cohomology sheaf $\mathbf{IC}(\mathfrak{M}_{0,0}(G))$ admits canonically the structure of a ring object in the bounded derived category of $\mathfrak{M}_{0,0}(G)$, and there is a natural ring isomorphism

$$I\!H^*(\mathfrak{M}_{0,0}(G))\cong I\!H^*(\mathfrak{M}_{0,0}(T))^W\big/\operatorname{Ann}(e_0).$$

This conjecture combines the preceding nonequivariant abelianization conjecture with the source's intersection-cohomology ring theorem, aiming to equip the intersection cohomology of the singular nonabelian quotient with a canonical ring structure.